The Stacks project

Exercise 111.52.1. Let $X$ be a Noetherian scheme. Let $\mathcal{F}$ be a coherent sheaf on $X$. Let $x \in X$ be a point. Assume that $\text{Supp}(\mathcal{F}) = \{ x \}$.

1. Show that $x$ is a closed point of $X$.

2. Show that $H^0(X, \mathcal{F})$ is not zero.

3. Show that $\mathcal{F}$ is generated by global sections.

4. Show that $H^ p(X, \mathcal{F}) = 0$ for $p > 0$.